Fig. 3: Twistronics without a twist: the case of a synthetic bilayer.

a A suitable two-dimensional Fermi gas (e.g. ⁸⁷Sr) with four distinct magnetic sub-levels chosen from the ground state manifold labelled by a pair of two-valued quantum numbers {σ, m} is trapped in a single layer state independent optical lattice, chosen here as a square lattice. This system forms a synthetic bilayer if one of the quantum numbers, say, m = ± 1/2 is identified with the layer degree of freedom. The fermion spin degree of freedom within a synthetic layer is given by σ = ↑, ↓. Each fermion species can tunnel between sites of the optical lattice with hopping parameter t. Additional Raman coupling Ω₀ can be utilized to induce transitions between m = + 1/2 and m = − 1/2 states effectively introducing (in general, tunable complex valued) interlayer hopping between the synthetic layers. An appropriate scheme utilzing a spatial light modulator can be used to engineer spatially modulated Raman coupling Ω(x, y) leading to systems with Moiré unit cell patterns. Panel b shows the synthetic bilayer obtained with \(\Omega (x,y)={\Omega }_{0}\left[1-\alpha (1+\cos (2\pi x/{l}_{x})\cos (2\pi y/{l}_{y}))\right]\), where l_x and l_y represent the periodicities along the x and y axes, respectively. c Tunable quasi-flat bands and Dirac cone spectra appear for special choices of periodicities. Shown here are bandstructures for (l_x, l_y) = (4, 4). Upper plot represents the negative part of the spectrum along the high symmetry points (we omit the postive part which is symmetric with respect to E = 0) for Ω₀α = 2t, while lower plot depicts strong bands flattening at Ω₀α = 20t. Figure adapted from ref. ⁵⁵ with permission of the American Physical Society.